Abstract : In this Manuscript, we investigate the influence of geometry on the Hardy-Sobolev equations on the compact Riemannian manifolds without boundary of dimension n > 2. More precisely, we prove in the non perturbative case that the existence of solutions depends only on the local geometry around the singularity when n > 3 while it is the global geometry of the manifold when n = 3 that matters. In the presence of a perturbative subcritical term, we prove that the existence of solutions depends only on the perturbation when n > 3 while an interaction between the perturbation and the global geometry appears in dimension 3.Finally, we establish an Optimal Hardy-Sobolev inequality for all compact Riemannian manifolds, with or without boundary, where we prove that the Riemannian sharp constant is the one for the Euclidean inequality and is achieved.